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4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.

4.97 Suppose that the positive random variables X₁,..., X, are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function k-1 rk (x). In particular, conclude that V has an exponential distribution with parameter k-1 Ak when each of the X; is exponen- tially distributed with parameter ₁.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.CR: Chapter 13 Review
Problem 61CR
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Question number 4.97

ery on al ne 0, 01 4.11 Probability Distributions and Entropy 205 4.97 Suppose that the positive random variables X₁,..., Xn are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function 1 rk(x). In particular, conclude that V has an exponential distribution with parameter 1 Ak when each of the X; is exponen- tially distributed with parameter λi. = 4.98 Suppose that the positive random variable X has the failure-rate function (1 + x)−¹ for x > 0. Verify the remarkable result that the reliability function of X is also given by (1+x)-¹ for x > 0. 4.99 The lifetime X of a semiconductor has the failure-rate function r(x) = λax-¹/(1+x) for x ≥ 0, where the parameters a and λ are positive. 2016 THE What is the reliabilitu of the lifetime X? Verify that the failure-
Transcribed Image Text:ery on al ne 0, 01 4.11 Probability Distributions and Entropy 205 4.97 Suppose that the positive random variables X₁,..., Xn are independent of each other and have respective failure-rate functions r₁(x),...,n(x). Prove that the random variable V = min(X₁,..., Xn) has the failure-rate function 1 rk(x). In particular, conclude that V has an exponential distribution with parameter 1 Ak when each of the X; is exponen- tially distributed with parameter λi. = 4.98 Suppose that the positive random variable X has the failure-rate function (1 + x)−¹ for x > 0. Verify the remarkable result that the reliability function of X is also given by (1+x)-¹ for x > 0. 4.99 The lifetime X of a semiconductor has the failure-rate function r(x) = λax-¹/(1+x) for x ≥ 0, where the parameters a and λ are positive. 2016 THE What is the reliabilitu of the lifetime X? Verify that the failure-
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